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La conjecture de Green générique

Arnaud Beauville (2003/2004)

Séminaire Bourbaki

Une courbe C projective et lisse de genre g , non hyperelliptique, admet un plongement canonique dans un espace projectif ℙ g - 1 . Un résultat classique affirme que l’idéal gradué I C des équations de C dans ℙ g - 1 est engendré par ses éléments de degré 2 , sauf si C admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de I C en fonction de l’existence de systèmes linéaires spéciaux sur C . Claire Voisin vient de...

Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves

Min Sha, Igor E. Shparlinski (2015)

Acta Arithmetica

We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of bounded height....

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Joseph H. Silverman (2012)

Journal de Théorie des Nombres de Bordeaux

A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to  1 modulo  m . We prove a similar result for polynomials  f ( X ) that are divisible in  ( ℤ / m ℤ ) [ X ] by a polynomial of the form 1 + X + ⋯ + X n for some n ≥ ϵ deg ( f ) . We also formulate and prove an analogous statement for elliptic curves.

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